![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sepnsepolem2 | Structured version Visualization version GIF version |
Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 47855. Proof could be shortened by 1 step using ssdisjdr 47792. (Contributed by Zhi Wang, 1-Sep-2024.) |
Ref | Expression |
---|---|
sepnsepolem2.1 | β’ (π β π½ β Top) |
Ref | Expression |
---|---|
sepnsepolem2 | β’ (π β (βπ¦ β ((neiβπ½)βπ·)(π₯ β© π¦) = β β βπ¦ β π½ (π· β π¦ β§ (π₯ β© π¦) = β ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sepnsepolem2.1 | . 2 β’ (π β π½ β Top) | |
2 | id 22 | . . 3 β’ (π½ β Top β π½ β Top) | |
3 | sslin 4230 | . . . . 5 β’ (π§ β π¦ β (π₯ β© π§) β (π₯ β© π¦)) | |
4 | sseq0 4395 | . . . . . 6 β’ (((π₯ β© π§) β (π₯ β© π¦) β§ (π₯ β© π¦) = β ) β (π₯ β© π§) = β ) | |
5 | 4 | ex 412 | . . . . 5 β’ ((π₯ β© π§) β (π₯ β© π¦) β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
6 | 3, 5 | syl 17 | . . . 4 β’ (π§ β π¦ β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
7 | 6 | adantl 481 | . . 3 β’ ((π½ β Top β§ π§ β π¦) β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
8 | ineq2 4202 | . . . . 5 β’ (π¦ = π§ β (π₯ β© π¦) = (π₯ β© π§)) | |
9 | 8 | eqeq1d 2729 | . . . 4 β’ (π¦ = π§ β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
10 | 9 | adantl 481 | . . 3 β’ ((π½ β Top β§ π¦ = π§) β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
11 | 2, 7, 10 | opnneieqv 47842 | . 2 β’ (π½ β Top β (βπ¦ β ((neiβπ½)βπ·)(π₯ β© π¦) = β β βπ¦ β π½ (π· β π¦ β§ (π₯ β© π¦) = β ))) |
12 | 1, 11 | syl 17 | 1 β’ (π β (βπ¦ β ((neiβπ½)βπ·)(π₯ β© π¦) = β β βπ¦ β π½ (π· β π¦ β§ (π₯ β© π¦) = β ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βwrex 3065 β© cin 3943 β wss 3944 β c0 4318 βcfv 6542 Topctop 22769 neicnei 22975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-top 22770 df-nei 22976 |
This theorem is referenced by: sepnsepo 47855 |
Copyright terms: Public domain | W3C validator |